If the letters of the word are arranged as inn a dictionary. M and n are the rank of the words BULBUL and NANNU respectively, then the value of m-4n is

Explanation

To solve the problem, we need to find the ranks of the words "BULBUL" and "NANNU" when all their letters are arranged in alphabetical order. Then we will calculate the value of \( m - 4n \), where \( m \) is the rank of "BULBUL" and \( n \) is the rank of "NANNU".

### Step 1: Find the rank of the word "BULBUL"

1. **Identify the letters and their frequencies**:
- The letters in "BULBUL" are B, U, L, B, U, L.
- Frequencies: B = 2, U = 2, L = 2.

2. **Arrange the letters in alphabetical order**:
- The alphabetical order of the letters is B, B, L, L, U, U.

3. **Calculate the total permutations of the letters**:
- The total permutations of the letters can be calculated using the formula for permutations of multiset:
\[ \text{Total permutations} = \frac{n!}{n_1! \cdot n_2! \cdot n_3!} = \frac{6!}{2! \cdot 2! \cdot 2!} = \frac{720}{8} = 90. \]

4. **Count the words before "BULBUL"**:
- Start with the first letter B:
- Fix B and arrange the remaining letters (B, L, L, U, U):
\[ \text{Permutations} = \frac{5!}{1! \cdot 2! \cdot 2!} = \frac{120}{4} = 30. \]
- Next, fix the second letter B:
- Now we have L, L, U, U left:
\[ \text{Permutations} = \frac{4!}{2! \cdot 2!} = \frac{24}{4} = 6. \]
- Fix the first letter L (which comes before U):
- We can arrange B, B, U, U:
\[ \text{Permutations} = \frac{4!}{2! \cdot 2!} = 6. \]
- Fix the first letter U (which comes after B):
- We can arrange B, B, L, L:
\[ \text{Permutations} = \frac{4!}{2! \cdot 2!} = 6. \]

5. **Calculate the rank of "BULBUL"**:
- Total words before "BULBUL":
\[ 30 + 6 + 6 = 42. \]
- So, the rank \( m \) of "BULBUL" is:
\[ m = 42 + 1 = 43. \]

### Step 2: Find the rank of the word "NANNU"

1. **Identify the letters and their frequencies**:
- The letters in "NANNU" are N, A, N, N, U.
- Frequencies: N = 3, A = 1, U = 1.

2. **Arrange the letters in alphabetical order**:
- The alphabetical order of the letters is A, N, N, N, U.

3. **Calculate the total permutations of the letters**:
- The total permutations of the letters can be calculated using the formula for permutations of multiset:
\[ \text{Total permutations} = \frac{5!}{3! \cdot 1! \cdot 1!} = \frac{120}{6} = 20. \]

4. **Count the words before "NANNU"**:
- Start with the first letter A:
- Fix A and arrange the remaining letters (N, N, N, U):
\[ \text{Permutations} = \frac{4!}{3! \cdot 1!} = 4. \]
- Fix the first letter N:
- Now we have A, N, N, U left:
\[ \text{Permutations} = \frac{4!}{2! \cdot 1! \cdot 1!} = 12. \]

5. **Calculate the rank of "NANNU"**:
- Total words before "NANNU":
\[ 4 + 12 = 16. \]
- So, the rank \( n \) of "NANNU" is:
\[ n = 16 + 1 = 17. \]

### Step 3: Calculate \( m - 4n \)

Now that we have \( m = 43 \) and \( n = 17 \):
\[ m - 4n = 43 - 4 \times 17 = 43 - 68 = -25. \]

### Final Answer
The value of \( m - 4n \) is \(-25\).

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